Standard tracking radar operates with a narrow beam to determine target bearing, and measured time of flight to determine distance. These techniques offer the advantages of direct measurement of target location and high antenna gain; but come with the associated cost of mechanically steered or phased-array antennas and substantial signal processing at the radar frequency. By solving the tracking problem using Doppler signals, with no direct bearing or range information, stationary, wide-beam antennas can be used and the signal processing can be done at the Doppler signal frequency.
The Doppler signal is generated when a moving body reflects waves originating with a transmitter. The reflected waves are received and combined with the transmitted signal in a homodyne detector, as seen in FIG. 1. The distance traveled by the wave is the sum of R.sub.1 (t), the distance from transmitter and target, and R.sub.2 (t), the distance from target to receiver. As the output of the homodyne detector, the Doppler signal is given by: ##EQU1## where A represents the amplitude, which is the consequence of the amplitudes of the transmitted and received signals, and the homodyne detector gain, .lambda. is the wavelength of the wave, and .theta..sub.o as the phase delay of the system.
The Doppler frequency shift can be observed with either a transmitter/receiver combination, and as seen in FIG. 1, or a signal which is emitted by the target and a receiver, as is the case for passive Doppler tracking, such as is described by Weinstein and Levanon, in their article "Passive array tracking of a continuous wave transmitting projectile," which was published on page 721 of volume 16, number 5 of the IEEE Transactions on Aerospace and Electronic Systems. Detection of the Doppler signal, however, requires the availability of the transmitted signal for homodyne detection, and thus is infeasible in most passive tracking applications. But when active systems, incorporating both transmitter and receiver, are used, it is straight forward to apply homodyne detection of the Doppler signal. The important advantage is that the Doppler signal includes phase information, which relates to the range to the target, as shown in equation (1).
Tracking using Doppler signals is accomplished by observing the Doppler signal at multiple points in space. An array of sensors and a moving target is illustrated in FIG. 2. The range to the i.sup.th sensor is given by: EQU R.sub.i (t)=.vertline.X(t)-Xs.sub.i .vertline. (2)
where X(t) is the target position expressed as a vector in Cartesian coordinates and Xs.sub.i is the location of the i.sup.th sensor. When the transmitter and receiver are collocated, as seen in FIG. 3, the distance travelled by the transmitted and received signals are approximately the same, and R.sub.1 (t).congruent.R.sub.2 (t). In this case, the Doppler shift frequency is given by: ##EQU2## where .phi..sub.i (t) is the angle between the target velocity vector and line of sight to the sensor, as seen in FIG. 2, and .DELTA.w.sub.i, is a frequency shift that may arise if homodyne detection is not used.
For the collocated transmitter and receiver, the Doppler signal is given by: ##EQU3## where the amplitude, A.sub.i, is a consequence of received and mixer signal amplitudes and filter gain; and where .theta..sub.o, is the phase delay of the system.
To illustrate the process of signal generation, the signals arising with a specific motion are seen in FIG. 4. The range between the target and the radar is seen in FIG. 4(A). It reaches its minimum at the point of closest approach. The received signal is seen in FIG. 4(B); in this simplified representation, constant amplitude is shown. The Doppler signal frequency is presented in FIG. 4(C), and is seen to be changing as the target/sensor geometry changes.
Because of the spherical symmetry of the Doppler process, a single sensor is inadequate to determine all of the parameters of a 3D trajectory. But when multiple sensors are used, the signals detected by the several sensors will be slightly different. When 3 or more sensors are used, only one trajectory will in general give rise to the signals detected at all of the sensors. Such a case is seen in FIG. 5, where the phase and frequency signals of 4 radars are presented. As seen in FIG. 5, both the phase and frequency of the Doppler signal show differences between the radars. To analyze the phase angle of the Doppler signal refers to determining the motion of the target based on the detected phase shift between the transmitted and received signals in active Doppler radars at several points in space. Correspondingly, to analyze the frequency of the Doppler signal refers to determining the motion of the target based on the detected Doppler shift frequency. In both cases, it is the differences between signals obtained at different points in space which permits the unique determination of the target motion.
The largest area of application of Doppler-based tracking has been passive tracking of a target which is emitting a sonic or electromagnetic signal. Descriptions of the basic operating principles are given by Chan and Jardine, in their article "Target localization and tracking from Doppler-shift Measurements", which was published on page 163 of volume 8, number 3 of the IEEE Journal of Oceanic Engineering; and by Weinstein and Levanon, in their article "Passive array tracking of a continuous wave transmitting projectile", which was mentioned above.
Recent work has shown that phase-based analysis improves performance in applications where the ratio of velocity at the point of closest approach, V.sub.pca, and distance to the point of closest approach, R.sub.pca, is large. The signal to noise ratio is proportional to .epsilon., which is given by: ##EQU4## where .lambda. is the received signal wavelength. For typical ranges and velocities, .epsilon. varies between 10.sup.4 for sonar applications, such as that described by Chart and Jardine; 10.sup.1 for the security application described in U.S. Pat. No. 4,236,140; 10.sup.0 for passive projectile tracking, such as that described by Weinstein and Levanon; 10.sup.-3 for applications to tracking for sports training (e.g., a baseball); and 10.sup.-4 for tracking small arms fire, as described by Colliver and Holcroft, in their article "Radar hostile fire location", which was published on page 32 of the Proceedings of the 1980 IEEE International Radar Conference.
The term .epsilon. indicates the ratios of the Cramer-Rao bounds on the signal to noise ratios of phase- and frequency-based 3D tracking. A small value, as for the sports training and small arms fire applications, indicates the use of phase-based tracking. Perhaps because the passive applications in sonar and projectile favor frequency-based tracking, all prior work in 3D tracking with Doppler signals has employed frequency-based methods.